decompwlj 3D

a(n) = 1^2 + 3^2 + 5^2 + 7^2 + ... + (2*n-1)^2 = n*(4*n^2 - 1)/3

A000447 on the OEIS · family polynomial · also known as Sums of the first n odd squares

Weight–level plate of a(n) = 1^2 + 3^2 + 5^2 + 7^2 + ... + (2*n-1)^2 = n*(4*n^2 - 1)/3
Click the plate to explore it in 3-D. Weight k across, level L up, both on log scales: blue in the weight class (k ≤ L), orange in the level class (k > L). The dashed diagonal is k = L.
Open in the 3-D viewerA000447 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,993
Level class, k > L99,993 · 100.00 %
Weight class, k ≤ L0 · 0.00 %
Ties, k = L0
On the level line L = 12,963
Forced level, l ≤ d²99,993
Range of a(n)0 … 1,333,293,333,699,999
Range of the jump d1 … 39,999,600,001
Largest weight k, level L1,327,541,738,436,367, 33,224

1^2 + 3^2 + ... + (2n-1)^2 = n(2n-1)(2n+1)/3. A cubic sequence. Every decomposable term is forced level (l <= d^2).

The decomposition

Every term of a strictly increasing sequence is written a(n) = k(n)·L(n) + d(n): the jump d = a(n+1) − a(n), the weight k the least divisor of a − d greater than d, the level L = (a − d)/k. A term decomposes when a > 2d; it is in the level class when k > L and in the weight class when k ≤ L. See decompwlj.com and arXiv:0711.0865, or how it works, with worked examples and a live one.